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### IMSL_SP_BDPDFAC

IMSL_SP_BDPDFAC

The IMSL_SP_BDPDFAC function computes the RTR Cholesky factorization of symmetric positive definite matrix, A, in band symmetric storage mode.

The IMSL_SP_BDPDFAC function computes the RTR Cholesky factorization of A. R is an upper triangular band matrix.

The L1 condition number of A is computed using Higham’s modifications to Hager’s method, as given in Higham (1988). If the estimated condition number is greater than 1/ε (where ε is the machine precision), a warning message is issued. This indicates that very small changes in A may produce large changes in the solution x.

The IMSL_SP_BDPDFAC function fails if any submatrix of R is not positive definite or if R has a zero diagonal element. These errors occur only if A is very close to a singular matrix or to a matrix which is not positive definite.

The IMSL_SP_BDPDFAC function is partially based on the LINPACK subroutines CPBFA and SPBSL; see Dongarra et al. (1979).

## Example

Solve a system of linear equations Ax = b, using both IMSL_SP_BDPDFAC and IMSL_SP_BDPDSOL, where: `n = 4L`
`ncoda = 2L`
`a = DBLARR((ncoda+1)*n)`
`a(0:n-1) = [0, 0, -1, 1]`
`a(n:2L*n-1) = [0, 0, 2, -1]`
`a(2L*n:*) = [2, 4, 7, 3]`
`; Define A in band symmetric storage mode.`
`b = [6, -11, -11, 19]`
`factor = IMSL_SP_BDPDFAC(a, n, ncoda)`
`; Use IMSL_SP_BDPDFAC to compute the factorization.`
`x = IMSL_SP_BDPDSOL(b, ncoda, Factor=factor)`
`; Compute the solution`
`PM, x`
`  4.0000000`
`  -6.0000000`
`  2.0000000`
`  9.0000000`

## Syntax

Result = IMSL_SP_BDPDFAC(A, N, Ncoda [, CONDITION=variable] [, /DOUBLE])

## Return Value

An array of size (Ncoda + 1) x n containing the RTR factorization of A in band symmetric storage mode.

## Arguments

### A

Array of size (Ncoda + 1) x n containing the n x n banded coefficient matrix in band symmetric storage mode A(i, j). See “Band Storage Format” for a description of band symmetric storage mode.

### N

Number of rows in a.

### Ncoda

Number of upper codiagonals in a.

## Keywords

### CONDITION (optional)

Named variable into which an estimate of the L1 condition number is stored.

### DOUBLE (optional)

If present and nonzero, double precision is used.

## Version History

 6.4 Introduced